Oversimplified: Signals and Systems (1) – Introduction

Many people see ‘(Linear) Signals and Systems’ as a hard course the first time they take it. It’s true because the materials are presented in the wrong order, so almost everybody did it the hard way the first time.

The materials in the class are taken for granted (i.e. you are supposed to be not just very familiar, but very comfortable with it) before you march into communications, control, MRI, audio and image processing. So in order to not hold you up for the motivating applications, pretty much all schools choose to teach ‘Signals and Systems’ early on right after you’ve taken the very basic ‘Linear Circuits 101’.

In fact ‘Linear Circuits 101’ has little to do with preparing you for ‘Signals and Systems’: it’s just a sanity check to see if you can handle basic high school algebra. In reality you’ll need way more math to breeze through ‘Signals and Systems’. This is the entire source of misery. A few made it and those who don’t will hate it, then switch areas.

There are only 2 true hard prerequisites (i.e. you have to know them cold) to master Signals and Systems:

Complex number manipulations (no calculus).

Enough linear algebra to be comfortable with inner products

Calculus (or differential equations**) is not really important here other than the basic definitions and simple derivatives (say, used in less than 5% of the time). If you need integration-by-parts for a problem, you are doing it the very long, painful, hard way.

Complex number manipulations are usually taught at the beginning, but many students thought they ‘know’ complex numbers and downplayed it. They obviously paid the price as the course progresses.

What’s really missing in the standard curriculum is linear algebra and inner products. As I’ll show later in the series, all the transforms you will learn in the class are simply inner products written out explicitly. You can recall all the transforms precisely in a split second if you know inner products.

The beauty of ‘Signals and Systems’ is that there are so many ways to interpret the same idea. This also means there’s a good chance that you’ll get yourself into a strenuous path if you blindly brute force everything straight from the definition*. The goal of ‘Ideas Oversimplified: Signals and Systems‘ series is to help you see through the smoke and find the easiest ways through intuition, not hard work and long derivations.

* I’m not telling you to downplay the definitions. In fact, they are the most important thing to understand first THEN remember (NOT the other way round). There are ‘gotcha’ problems which the answer came straight off the definition that makes you look stupid once they tell you the answer. More on that later.

** Actually, one of the main punchlines of this class is that you are solving linear constant coefficient differential/difference equations (LCCDE) the hard way in differential equations class. Linear systems is basically a shorthand for LCCDE and all the tools you learn (convolution and transformation) are manipulating differential/difference equations under the hood. See Heaviside Operator Theory if you are interested.